Effect of Hunting on Red Deer

P15.2 Fortgeschrittenes Praxisprojekt

Nikolai German, Thomas Witzani, Ziqi Xu, Zhengchen Yuan, Baisu Zhou

Dr. Nicolas Ferry - Bavarian National Forest Park / Daniel Schlichting - StabLab

31 Jan 2025

Agenda

  1. The Background
  1. The Data
  1. The Models
  1. The Wrap-up

Motivation

  • Hunting activities have a numerical effect on animal populations
  • Additionally, hunting can have non-lethal effects
  • Goal: assess short-term stress response in red deer towards hunting events at the Bavarian Forest National Park

Data-Generating Process

  • A deer roams freely in the Bavarian Forest National Park
  • Its movement is tracked by a GPS collar
  • A hunting event happens
  • After some time, the deer defecates. The defecation event
  • Subsequently, Researchers go to the defecation location and collect a fecal sample

FCMs as a Measure of Stress

  • Faecal Cortisol Metabolites (FCM) are substances found in feces of animals
  • The FCM level is used to measure previous stress. Higher Stress \(\implies\) Higher FCM level
  • Stress \(\Rightarrow\) secretion of certain hormones \(\Rightarrow\) gut retention \(\Rightarrow\) FCM
  • Gut retention time \(\approx\) 19 hours
  • Once defecated, FCM levels decay over time

Huber et al (2003)

Research Questions

  • What is the effect of temporal and spatial distance on FCM levels?
  • Does the time between defecation event and sample collection effect FCM levels?

Approach

  • Model FCM levels - amongst other covariables - on spatial and temporal distance to hunting activities

  • Expectations:

    • FCM levels higher when closer in time and space
    • FCM levels lower, the more time passes between defecating and sampling

Agenda

  1. The Background
  1. The Data
  1. The Models
  1. The Wrap-up

The Datasets

  • FCM Data
  • Hunting Events
  • Movement Data

FCM Data

Contains information of 809 faecal samples, including:

  • the FCM level [ng/g]
  • the time and location of sampling
  • to which deer the sample belongs
  • when the defecation happened

Samples where taken at irregular time intervals from 2020 to 2022.

Hunting Events

Contains location and time of \(\geq\) 700 hunting events from 2020 to 2022.

  • 519 hunting events have complete location and time information.

Movement Data

  • Contains the location of the 40 collared deer from Feb. 2020 to Feb. 2023
  • Movement is tracked at hourly intervals

Limited Data, Large Uncertainty

  • Hunting events are single points in time and space.
  • Deer locations at hourly intervals \(\Rightarrow\) exact distances unknown \(\Rightarrow\) approximate needed, large uncertainty!
  • Each deer only encountered few hunting events.

Limited Data, Large Uncertainty

Other sources of uncertainty include:

  • unknown characteristics of the deer (e.g., age, health, etc.),

  • other unknown stressors (e.g., predators, human activities, weather, etc.),

  • unknown geographical features (e.g., terrain could affect the propagation of sound).

Distance Approximation

Deer location at the time of hunting event is approximated by linear interpolation:

Relevant Hunting Events

A hunting event is considered relevant to an FCM sample, if

  • the time difference between defecation and hunting is bounded within gut retention time (GRT) thresholds [hours], and
  • the spatial distance between deer and hunting is below a distance threshold [km].

In the following, GRT thresholds = (0, 50), distance threshold = 10 or 15.

The Most Relevant Hunting Event

Among the relevant hunting events, the most relevant one is defined by one of the three introduced proximity criteria:

  • the closest in time to GRT target = 19 hours (“closest in time”)
  • the closest in space (“nearest”)
  • the one with the “highest score

Illustration

TimeDiff Distance 19 hours distance threshold GRT highthreshold Number of otherrelevant huntingevent = 3 Deer Hunting events Nearest Highestscore Closestin time(to 19 hours)

A hunting event is considered relevant to a FCM sample, if

  • the time difference between experiencing stress (hunting) and defecation is between the GRT thresholds, and
  • the distance between the deer and the hunting event is \(\leq\) distance threshold.

The Scoring Function

we define the Scoring function as following:

\[ S(d, t) \propto \begin{cases} \frac{1}{d^2} \cdot f_\textbf{t}(t), t \sim \mathcal{N}(\mu, \sigma^2) &|t \leq \mu \\ \frac{1}{d^2} \cdot f_\textbf{t}(t), t \sim \mathcal{Laplace}(\mu, b) &|t > \mu \end{cases} \] where:

\[ \begin{align*} d & \text{: Distance } \\ t & \text{: Time Difference } \\ \mu & \text{: GRT target = 19 hours } \end{align*} \]

The Scoring Function

The marginal effects of distance and elapsed time since challenge on the score:

The Fused Data

Finish Datasets

We suggest three different Datasets for Modelling

DataSet GRT low GRT high Distance Threshold Proximity Criterion Deers Observations
1 0 36 10 closest in time 35 149
2 0 36 10 nearest 35 147
3 0 200 15 score 36 223

Agenda

  1. The Background
  1. The Data
  1. The Models
  1. The Wrap-up

The Models

For Modelling, we consider the following covariates, defined for each pair of FCM sample and most relevant hunting event:

  • Time Difference
  • Distance
  • Sample Delay
  • Defecation Day (as Day of Year (1-366))
  • Number of other relevant hunting events

The Models

We chose two different approaches to Modelling:

  1. Machine Learning: a model, which focuses on prediction, in our case a XGBoost Model
  2. Statistical Modelling: a model, which helps to understand the effects of our covariables, here a General Additive Mixed Model

A. XGBoost

TBD

B. Generalized Additive Mixed Model

  • Family: Gamma

  • Log link for interpretability

  • Let \(i = 1,\dots,N\) be the indices of deer and \(j = 1,\dots,n_i\) be the indices of FCM measurements for each deer

\[ \begin{eqnarray} \textup{FCM}_{ij} &\sim& \mathcal{Ga}\left( \nu, \frac{\nu}{\mu_{ij}} \right) \\ \mu_{ij} &=& \mathbb{E}(\textup{FCM}_{ij}) = \exp(\eta_{ij}) \\ \eta_{ij} &=& \beta_0 + \beta_1 \textup{Pregnant}_{ij} + \beta_2 \textup{NumberOtherHunts}_{ij} + \\ && f_1(\textup{TimeDiff}_{ij}) + f_2(\textup{Distance}_{ij}) + \\ && f_3(\textup{SampleDelay}_{ij}) + f_4(\textup{DefecationDay}_{ij}) + \\ && \gamma_{i}, \\ \gamma_i &\overset{\mathrm{iid}}{\sim}& \mathcal{N}(0, \sigma_\gamma^2). \end{eqnarray} \]

B Generalized Additive Mixed Model

Closest in Time

B Generalized Additive Mixed Model

Nearest

B Generalized Additive Mixed Model

Highest score

B Coefficient Table

Linear Effects:

Dataset Term Estimate Std. Error
Closest in Time (Intercept) 5.8243844 0.0533979
Closest in Time NumOtherHunts -0.1370438 0.0614158
Nearest (Intercept) 5.8123504 0.0541316
Nearest NumOtherHunts -0.1026115 0.0596574
Highest Score (Intercept) 5.8882327 0.0812529
Highest Score NumOtherHunts -0.0112701 0.0141569

Agenda

  1. The Background
  1. The Data
  1. The Models
  1. The Wrap-up

Conclusion

  • Due to the high uncertainties, we were not able to detect a relevant effect of spatial or temporal distance on FCM levels
  • In some of the cases we were able to prove the expected decay of FCM levels with prolonged time between defecation event and sample collection
  • With more datapoints, the uncertainty will likely shrink

Discussion

  • How to minimize spatial and temporal distance at the same time?

  • How to use a bigger Part of the Data?

Appendix

FCM Samples and Hunting Events Over Time